Aaron is correct. Don't damn a technology because somebody did a first implementation that didn't work well. If there are ground loops or an incorrect grounding scheme, lots of funky things can occur, and noise issues will be exacerbated.
It really depends a lot on how the machine is wired. Is the power or return for the solenoids running in the same bundle as the data signal wire for the LEDs? If those are separated or in different bundles, it seems like there shouldn't be many issues. Is the machine grounded properly and keeping the solenoid grounds separated from the signal/digital grounds. Is the grounding for the machine in a star formation to reduce the noise? Are the solenoids being driven by FETs or Darlingtons? (FETs have faster rising edges, so they induce more voltage on wires run in parallel).
All of the flickering issues using shift registers to drive LED bulbs that happened in my machine, have been caused by an impedence mismatch between the output driver, and the input receiver and having a long run of wire between them. The wire acts as an antenna and can amplify the edges because of reflected waves. I needed to add series terminating resistors on these lines. If a series LED strip is split, and the wires between the split are long (6 inches or more), a series terminating resistor may need to be added on the data signal wire.
My issues were not caused by induced voltage from kicking the solenoids. I ran the signal wires in the same bundle as the solenoid power, and then I separated the signal wires into a separate bundle and ran the same test. I saw extra clock edges in both cases. While separating the wires didn't make a difference in my case...it is definitely the right thing to do to separate solenoid power/grounds wires from digital power ground wires.
I will have more data on using the series LEDs in the next couple of months. Is a single series LED (or neopixel) bright enough to light up an insert, or will they need to be cut apart and doubled up to get it bright enough? Does the light need to be diffused, and if so, what should be used to diffuse the light to get an even insert lighting?